Misere solutions for particular octal games

October 25, 2006 at 5:40 am (algebra, solutions)

Some commutative monoids for specific games are now online for you to browse, if you like.

There’s more to come!

An invitation to combinatorial games

October 13, 2006 at 6:00 pm (lectures)

Aaron Siegel’s slides from his recent talk, “An invitation to combinatorial games, (Part I).”

The commutative algebra of T2

October 6, 2006 at 12:01 am (algebra, nim, papers)

I’ve been reading about semigroup rings and other things in commutative algebra. In looking though old notes I found this email from Aaron Siegel from about a year ago, recounting some interesting information from a meeting with David Eisenbud. It corresponds pretty closely to what I was trying to compute myself, this morning

The meeting went well. He showed me a new (to me) way of looking at monoids in terms of commutative rings. Namely, instead of considering the monoid

Q = {a,b : a^2=1,b^3=b }

[ Aside: the monoid Q is the (tame) misere quotient of the game of Nim played using heaps of size 1 and 2 only. In this paper, Q is called T2. ]

we instead look at the ring

R = k[a,b] / (1-a^2,b-b^3)

and now we can apply the techniques of commutative algebra to analyze this ring. One observation is that because Q six elements, R is a 6-dimensional vector space over k. A basis for this vector space is Q itself, but it’s instructive to look at other bases and to study various decompositions of R.

For example, if x is an idempotent of R, then so is 1-x, since (1-x)(1-x) = 1-2x+x^2 = 1-2x+x = 1-x. Further, x(1-x) = 0, so R can be written as the internal direct sum of R/(x) and R/(1-x), with the isomorphism

R -> R/(x) + R/(1-x)

given by y -> ((1-x)y,xy).

Using the above example, we have

R = R/(b^2) + R/(1-b^2),

and it’s not hard to see that R/(1-b^2) is four-dimensional with basis {b^2,ab^2,b,ab} (which you’ll recognize as a maximal subgroup of Q), and R/(b^2) is two-dimensional with basis {1-b^2,a-ab^2} (which of course is equal to {1,a} in R/(b^2)).

Note the parallel with semigroup theory. In the ring R/(1-b^2), we’ve asserted that b^2 = 1, making b^2 into the identity. So the group {b^2,ab^2,b,ab} corresponds exactly with the multiplicative structure of R/(1-b^2). In other words, R/(1-b^2) is isomorphic to the group ring k[Z_2 x Z_2]. Likewise, in R/(b^2) we’ve set b^2 = 0 and the result is isomorphic to the group ring k[Z_2], corresponding to the maximal subgroup {1,a}. So the ring decomposition by idempotents matches the monoid decomposition we’re used to (archimedean components).

All very interesting. Eisenbud also pointed out that the rings can be decomposed further:

1 – b^2 = (1-b)(1+b) so R/(1-b^2) = R/(1-b) + R/(1+b),

and we can further break off the “a” part of each component, since it never interacts with the “b” component. This fully decomposes R as the vector space product of six copies of k and gives a new basis.

Finally, we know that we always have a^2 = 1 in any misere quotient Q. Because of this, the corresponding ring R always decomposes into two isomorphic components

R = R/(1-a) + R/(1+a)

and it’s simpler, therefore, to discard a and study just one of these (R/(1-a) is preferable, since that’s equivalent to just setting a = 1.) Of course, the structure of the P-positions is different on each component. It’s unclear what all of this might mean in game-theoretic terms.

He also recommended a book by Gilmer, “Commutative Semigroup Rings,” which I’ve checked out from Berkeley’s library. The paper by Eisenbud & Sturmfelds, “Binomial Ideals,” might also be useful (it’s on the arxiv).

An invitation to combinatorial games

October 4, 2006 at 6:13 am (abstracts, lectures)

Aaron Siegel is giving talks on combinatorial games at the Institute for Advanced Study in Princeton, NJ this month, on Oct 10 and 17.

Here is his abstract in the announcement:

Combinatorial game theory is the study of combinations of two-player games with no hidden information and no chance elements. The subject has its roots in recreational mathematics, but in its modern form involves a rich interplay of ideas borrowed from algebra, combinatorics, and the theory of computation.

The first part of this talk will be a general introduction to the classical theory of partizan games. I will show how a few simple axioms give rise to the group of short games, a partially-ordered Abelian group with enormously rich structure. I will discuss how the theory can be applied to extract useful information about a diverse array of games, including Nim, Domineering, Go, and to a lesser extent Chess.

In the second half of the talk, I will briefly discuss three areas of current research in combinatorial games: the theory of misere quotients; the lattice structure of short games; and the temperature theory of Go endgames. There has been significant activity in all three topics in the last five years, but nonetheless I will be able to state some major (and reasonably elementary) open problems in each of them.

The Phi-values of various games

October 2, 2006 at 3:29 am (papers, software)

Just in time for Richard K. Guy’s 90th birthday a few days ago (30 September 2006), Aaron Siegel and I finished our paper The Phi-values of various games.

It just appeared on the arXiv.  Check it out!